材料力学闭口薄壁截面剪应力和单位扭转角公式推导
剪应力公式：
![](https://wkrtcs.bdimg.com/rtcs/image?w=77.33&md5sum=970e748d7d2cc550479438120e7fac63&sign=2e2f0c0ef9&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=51&ipr={"t":"img","r":"r_2","w":"77.33","h":"36.00","dataType":"gif","c":"word/media/image1.gif"})
单位扭转角公式：
![](https://wkrtcs.bdimg.com/rtcs/image?w=129.33&md5sum=970e748d7d2cc550479438120e7fac63&sign=2e2f0c0ef9&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=51&ipr={"t":"img","r":"r_2","w":"129.33","h":"52.00","dataType":"gif","c":"word/media/image2.gif"})
1、从薄壁杆中取微段dx,在微段上去单元体abcd,设截面a点处的切应力为![](https://wkrtcs.bdimg.com/rtcs/image?w=30.67&md5sum=970e748d7d2cc550479438120e7fac63&sign=2e2f0c0ef9&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=51&ipr={"t":"img","r":"r_3","w":"30.67","h":"13.33","dataType":"gif","c":"word/media/image3.gif"})
，壁厚为；b点切应力为，壁厚为。根据切应力互等定理，在纵向截面ad和bc截面上的切应力分别为和，剪力分别为和，有平衡条件
![](https://wkrtcs.bdimg.com/rtcs/image?w=190.67&md5sum=970e748d7d2cc550479438120e7fac63&sign=2e2f0c0ef9&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=51&ipr={"t":"img","r":"r_2","w":"190.67","h":"36.00","dataType":"gif","c":"word/media/image9.gif"})
得
![](https://wkrtcs.bdimg.com/rtcs/image?w=81.33&md5sum=970e748d7d2cc550479438120e7fac63&sign=2e2f0c0ef9&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=51&ipr={"t":"img","r":"r_2","w":"81.33","h":"16.00","dataType":"gif","c":"word/media/image10.gif"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=342.13&md5sum=970e748d7d2cc550479438120e7fac63&sign=2e2f0c0ef9&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=51&ipr={"t":"img","r":"r_4","w":"342.13","h":"553.73","dataType":"jpeg","c":"word/media/image11.jpeg","imgOriW":784,"imgOriH":1269})
由于a和b是横截面上任意点，这说明在横截面上的任意点，切应力与该点处壁厚的乘积为一常数，即
![](https://wkrtcs.bdimg.com/rtcs/image?w=74.67&md5sum=970e748d7d2cc550479438120e7fac63&sign=2e2f0c0ef9&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=51&ipr={"t":"img","r":"r_2","w":"74.67","h":"16.00","dataType":"gif","c":"word/media/image12.gif"})
在图（c）面积微元![](https://wkrtcs.bdimg.com/rtcs/image?w=38.67&md5sum=970e748d7d2cc550479438120e7fac63&sign=2e2f0c0ef9&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=51&ipr={"t":"img","r":"r_2","w":"38.67","h":"16.00","dataType":"gif","c":"word/media/image13.gif"})
上的微剪力为,他对横截面内任意点O的矩为，为O点到面积微单元所在截面中心线切线的垂直距离。沿截面中心线积分得
![](https://wkrtcs.bdimg.com/rtcs/image?w=198.67&md5sum=970e748d7d2cc550479438120e7fac63&sign=2e2f0c0ef9&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=51&ipr={"t":"img","r":"r_2","w":"198.67","h":"50.67","dataType":"gif","c":"word/media/image17.gif"})
式中![](https://wkrtcs.bdimg.com/rtcs/image?w=41.33&md5sum=970e748d7d2cc550479438120e7fac63&sign=2e2f0c0ef9&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=51&ipr={"t":"img","r":"r_3","w":"41.33","h":"14.67","dataType":"gif","c":"word/media/image18.gif"})
为图中三角形OAB面积的两倍，所以积分是截面中线所围面积的两倍，即
![](https://wkrtcs.bdimg.com/rtcs/image?w=84.00&md5sum=970e748d7d2cc550479438120e7fac63&sign=2e2f0c0ef9&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=51&ipr={"t":"img","r":"r_2","w":"84.00","h":"14.67","dataType":"gif","c":"word/media/image21.gif"})
于是可求得
此即闭口薄壁截面杆扭转时，横截面上任意点的切应力公式。
2、由圆轴扭转可得，相对扭转角公式：
![](https://wkrtcs.bdimg.com/rtcs/image?w=124.00&md5sum=970e748d7d2cc550479438120e7fac63&sign=2e2f0c0ef9&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=51&ipr={"t":"img","r":"r_2","w":"124.00","h":"36.00","dataType":"gif","c":"word/media/image22.gif"})
注意：这里的![](https://wkrtcs.bdimg.com/rtcs/image?w=26.67&md5sum=970e748d7d2cc550479438120e7fac63&sign=2e2f0c0ef9&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=51&ipr={"t":"img","r":"r_6","w":"26.67","h":"13.33","dataType":"gif","c":"word/media/image23.gif"})
是一个变量
可得
![](https://wkrtcs.bdimg.com/rtcs/image?w=62.67&md5sum=970e748d7d2cc550479438120e7fac63&sign=2e2f0c0ef9&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=51&ipr={"t":"img","r":"r_2","w":"62.67","h":"36.00","dataType":"gif","c":"word/media/image24.gif"})
所以
![](https://wkrtcs.bdimg.com/rtcs/image?w=397.33&md5sum=970e748d7d2cc550479438120e7fac63&sign=2e2f0c0ef9&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=51&ipr={"t":"img","r":"r_2","w":"397.33","h":"53.33","dataType":"gif","c":"word/media/image25.gif"})
故
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